Shape Convergence for Aggregate Tiles in Conformal Tilings

Abstract

Given a substitution tiling $T$ of the plane with subdivision operator $\tau$, we study the conformal tilings ${\mathcal{T}}_n$ associated with ${\tau }^{n}T$. We prove that aggregate tiles within ${\mathcal{T}}_n$ converge in shape as $n\to \infty$ to their associated Euclidean tiles in $T$.